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Discrete-Time Pairwise Connected Switched Systems and Lur’e Systems. Tsypkin’s Criterion for Systems with Two Nonlinearities

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Abstract

We study the stability of discrete-time switched systems for any laws of switching between linear subsystems. Pairwise connected systems are distinguished among such systems. A sufficient frequency-domain stability condition has been obtained for them. Two sufficient conditions and two criteria for the existence of a Lyapunov quadratic function are obtained for switched systems whose stability is equivalent to the absolute stability of Lur’e systems with two nonlinearities. These conditions amount to checking the solvability of special matrix inequalities whose dimensions are considerably lower than the dimension of the original system of matrix inequalities that defines the necessary and sufficient conditions. The resulting conditions are compared with the conditions of the Tsypkin criterion and with the necessary and sufficient conditions using the examples of systems of the third and sixth orders.

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REFERENCES

  1. Shorten, R., Wirth, F., Mason, O., et al., Stability сriteria for switched and hybrid systems, SIAM Rev., 2007, no. 4, pp. 545–592.

  2. Lin, H. and Antsaklis, P.J., Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Autom. Control, 2009, vol. 54, no. 2, pp. 308–322.

    Article  MathSciNet  MATH  Google Scholar 

  3. Fradkov, A., Early ideas of the absolute stability theory, 2020 Eur. Control Conf. (ECC) (St. Petersburg, May 12–15, 2020), pp. 762–768. .

  4. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Linear matrix inequalities in control systems with uncertainty, Autom. Remote Control, 2021, vol. 82, no. 1, pp. 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  5. Aleksandrov, A.Yu. and Platonov, A.V., On stability of solutions for a class of nonlinear difference systems with switching, Autom. Remote Control, 2016, vol. 77, no. 5, pp. 779–788.

    Article  MathSciNet  MATH  Google Scholar 

  6. Proskurnikov, A.V. and Matveev, A.S., Tsypkin and Jury-Lee criteria for synchronization and stability of discrete-time multiagent systems, Autom. Remote Control, 2018, vol. 79, no. 6, pp. 1057–1073.

    Article  MathSciNet  MATH  Google Scholar 

  7. Kamenetskiy, V.A., Frequency-domain stability conditions for discrete-time switched systems, Autom. Remote Control, 2018, vol. 79, no. 8, pp. 1371–1389.

    Article  MathSciNet  MATH  Google Scholar 

  8. Kamenetskiy, V.A., Switched systems, Lur’e systems, absolute stability, Aizerman problem, Autom. Remote Control, 2019, vol. 80, no. 8, pp. 1375–1389.

    Article  MathSciNet  MATH  Google Scholar 

  9. Gusev, S.V. and Likhtarnikov, A.L., Kalman–Popov–Yakubovich lemma and the \(S \)-procedure: a historical essay, Autom. Remote Control, 2006, vol. 67, no. 11, pp. 1768–1810.

    Article  MathSciNet  MATH  Google Scholar 

  10. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems under Exogenous Disturbances: Technique of Linear Matrix Inequalities), Moscow: LENAND, 2014.

    Google Scholar 

  11. Kamenetskii, V.A., Absolute stability of discrete control systems with nonstationary nonlinearities, Avtom. Telemekh., 1985, no. 8, pp. 172–176.

  12. Yakubovich, V.A., Absolute stability of pulsed systems with several nonlinear or linear but nonstationary blocks. I, II, Autom. Remote Control, 1967, vol. 28, no. 9, pp. 1301–1313; 1968, vol. 29, no. 2, pp. 244–263.

  13. Shepel’yavi, A.I., Absolute instability of nonlinear pulse-amplitude control systems. Frequency criteria, Autom. Remote Control, 1972, vol. 33, no. 6, pp. 929–935.

    Google Scholar 

  14. Kamenetskiy, V.A., Frequency-domain stability conditions for hybrid systems, Autom. Remote Control, 2017, vol. 78, no. 12, pp. 2101–2119.

    Article  MathSciNet  MATH  Google Scholar 

  15. Molchanov, A.P., Lyapunov functions for nonlinear discrete-time control systems, Autom. Remote Control, 1987, vol. 48, no. 6, pp. 728–736.

    MATH  Google Scholar 

  16. Kamenetskiy, V., Stability conditions for systems with switching between four linear discrete subsystems, Proc. 15th Int. Conf. “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskii Conf.) (STAB-2020) (Moscow), New York: IEEE Catalog no. CFP20E79-ART, 2020, pp. 1–4. .

  17. Balandin, D.V. and Kogan, M.M., Sintez zakonov upravleniya na osnove lineinykh matrichnykh neravenstv (Synthesis of Control Laws Based on Linear Matrix Inequalities), Moscow: Fizmatlit, 2007.

    Google Scholar 

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Funding

This work was supported by the Program of Fundamental Scientific Research in Priority Areas determined by the Presidium of the Russian Academy of Sciences, no. 7 “New developments in promising areas of energy, mechanics, and robotics.”

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  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997, Russia

    V. A. Kamenetskiy

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  1. V. A. Kamenetskiy
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Correspondence to V. A. Kamenetskiy.

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Translated by V. Potapchouck

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Kamenetskiy, V.A. Discrete-Time Pairwise Connected Switched Systems and Lur’e Systems. Tsypkin’s Criterion for Systems with Two Nonlinearities. Autom Remote Control 83, 1371–1392 (2022). https://doi.org/10.1134/S000511792209003X

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  • DOI: https://doi.org/10.1134/S000511792209003X

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